The onset of zonal modes in two-dimensional Rayleigh-Bénard convection

We study the stability of steady convection rolls in 2D Rayleigh-Bénard convection with free-slip boundaries and horizontal periodicity over twelve orders of magnitude in the Prandtl number (10^-6 ≤ Pr ≤ 10⁶) and five orders of magnitude in the Rayleigh number (8π⁴ < Ra ≤ 3 × 10⁷). The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio Γ = 2, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the (Pr,Ra)-plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit Pr → 0 and find that the steady convection rolls become unstable almost instantaneously, eventually leading to non-linear relaxation osculations and bursts, which we can explain with a weakly non-linear analysis. In the complementary large-Pr limit, we observe that the stability boundary reaches an asymptotic value Ra = 2.54 × 10⁷ and that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.